1. Introduction to Problem Solving
Psychology 355: Cognitive Psychology Instructor: John Miyamoto 05/26/2015: Lecture 09-2
There is no Lecture 09-1 because Monday was Memorial Day (no lecture on that day.)
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2. Outline Definition of “problem”
Information processing versus Gestalt approach to problem solving. Algorithmic problems & insight problems
Tower of Hanoi – an example of an algorithmic problem
Insight problems
Problem representation
Problem restructuring
Problem isomorphs
Definition of Problem Solving
Psych 355, Miyamoto, Spr '15

3. Definition of Problem Solving
A problem exists when the present state differs from a goal state. The problem is to change the present state into the goal state.
Initial state
Goal state
Permissible "moves" – ways to change the problem state from the initial state towards the goal state.
Interesting problems are situations where it is not obvious how to change the initial state into the goal state.
Cognitive psychology of problem solving – how do people solve problems.
Examples of Problem Solving Situations
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4. Problem Solving - Examples
Math problems, physics problems, science problems generally.
Initial state: The given information in the problem.
Goal state: The “answer” or solution to the problem.
Practical problems, e.g., arranging furniture, building a mechanical device.
Winning strategies in games, business, public health, law & war.
Key Ideas in Theory of Problem Solving
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5. Key Ideas in the Psychology of Problem Solving
Problem representation – The mental representation of the problem that the problem solver manipulates while trying to solve the problem.
Initial state
Goal state
Moves or transformations. Constraints and rules.
Insight problems & algorithmic problems
Restructuring a problem representation
Set
Functional fixedness
Algorithmic vs Insight Problems
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6. Algorithmic Problems versus Insight Problems
Algorithmic problems: The initial problem state can be transformed to the goal state by a systematic procedure.
Example: The Tower of Hanoi
Example: Solving a long division problem
Insight problems require mental restructuring of the problem representation to get a solution.
Circle problem
Mutilated checkerboard problem
Algorithmic and insight problems require somewhat different psychological processes to solve them.
Tower of Hanoi – Example of an Algorithmic Problem
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7. The Tower of Hanoi (A Problem with an Algorithmic Solution)
We will discuss algorithmic problems tomorrow.
Long division is an example of an algorithmic problem
Multiplying two numbers is an algorithmic problem.
Finding the square root of a positive number is an algorithmic problem.
Tower of Hanoi is an algorithmic problem – there is a logically adequate strategy that will always solve this problem.
General Idea of an Insight Problem
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8. General Idea of an Insight Problem
The solution of insight problems usually depends on finding a new way to represent the problem.
Ideas from Gestalt Psychology
The mind searches for structure in perception
The mind searches for structure in problem solving
Mental Representation of a Problem
The Problem Representation =
Finding a New Way to Represent a Problem
Restructuring the Problem Representation =
Solving the Circle Problem by Restructuring the Problem Representation
Psych 355, Miyamoto, Spr '15

9. The Circle Problem: An Example of an Insight Problem
Given:
radius r = 1
length of a = 0.9
line b is perpendicular to line a
Question: What is the length of x?
Hint: Change the problem representation.
#Section:
plot.jm(x=c(-100, 100), y = c(-100, 120), axes=F)
ellipse(c(0,0), width=160, ht = 160, lwd=2)
lines(c(0,0), c(-80, 80), lwd=2)
lines(c(-80, 80), c(0,0), lwd=2)
tt <- pi/6
rr <- 80
aa <- -cos(tt)*rr
bb <- sin(tt)*rr
#lines(c(0, aa), c(0, bb), lwd=2)
lines(c(aa,aa), c(0, bb), lwd=2)
lines(c(aa, 0), c(bb,bb), lwd=2)
lines(c(aa, 0), c(0,bb), lwd=2)
text(x = aa + 5, y = bb/1.95, "a", cex=1.5)
text(x = -2 + aa/2, y = bb/2 - 7, "x", cex=1.5)
text(x = 40, y = -7, "r", cex=1.5)
text(x = c(-80), y = (95),
c(paste("r = 1.0, a = ", round(aa/80 , dig=1))), cex=2, adj=0)
text(x = c(-80), y = (120), "What is the length of x?", cex=2, adj=0)
#lines(c(0, aa), c(0, bb), lwd=2, lty=2)
#EndSection:
Initial Representation
Restructuring the Representation of the Circle Problem
Psych 355, Miyamoto, Spr '15

10. Restructuring the Representation of the Circle Problem
If r = 1, a = 0.9, and a and b are perpendicular, what is the length of x?
Solution: Add dashed line that connects the opposite corners.
Alternative representation: The answer is obvious: x = r = 1.
Alternative problem representation makes the solution obvious.
Solutions to insight problems often depend on a “trick”.
Here the trick is to change the problem representation.
#Section:
plot.jm(x=c(-100, 100), y = c(-100, 120), axes=F)
ellipse(c(0,0), width=160, ht = 160, lwd=2)
lines(c(0,0), c(-80, 80), lwd=2)
lines(c(-80, 80), c(0,0), lwd=2)
tt <- pi/6
rr <- 80
aa <- -cos(tt)*rr
bb <- sin(tt)*rr
#lines(c(0, aa), c(0, bb), lwd=2)
lines(c(aa,aa), c(0, bb), lwd=2)
lines(c(aa, 0), c(bb,bb), lwd=2)
lines(c(aa, 0), c(0,bb), lwd=2)
text(x = aa + 5, y = bb/1.95, "a", cex=1.5)
text(x = -2 + aa/2, y = bb/2 - 7, "x", cex=1.5)
text(x = 40, y = -7, "r", cex=1.5)
text(x = c(-80), y = (95),
c(paste("r = 1.0, a = ", round(aa/80 , dig=1))), cex=2, adj=0)
text(x = c(-80), y = (120), "What is the length of x?", cex=2, adj=0)
#lines(c(0, aa), c(0, bb), lwd=2, lty=2)
#EndSection:
Alternate Representation for the Circle Problem
Another Insight Problem – the Mutilated Checkerboard Problem
Psych 355, Miyamoto, Spr '15

11. Another Insight Problem – Mutilated Checkerboard Problem
Problem: Cover the mutilated checkerboard with domino pieces so that every domino covers two squares OR if this is impossible, explain why it is impossible. The domino pieces must always be perpendicular or parallel to the sides of the board - they cannot be placed in a diagonal position.
See ‘e:\p355\hnd10-1a.doc’ and ‘e:\p355\hnd10-1b.doc’ for code for making mutilated checkerboards.
#Section:
plot.jm(c(-1, 9), c(-1, 10), axes=F)
j.dark <- colors()[82]
j.light <- 8
for (i in 1:4) for (j in 1:8) {
II <- (i - 1)*2 + .5
JJ <- j - .5
if (j %in% c(1,3,5,7)) j.col <- j.light else j.col <- j.dark
if (i != 1 | j != 8) rectan(c(JJ,II), width=1, ht=1, col=j.col) }
II <- (i - 1)*
if (j %in% c(2,4,6,8)) j.col <- j.light else j.col <- j.dark
if (i != 4 | j != 1) rectan(c(JJ,II), width=1, ht=1, col=j.col)
lines(c(0,0), c(0, 7), lwd=3)
lines(c(0,1), c(7, 7), lwd=3)
lines(c(1,1), c(7, 8), lwd=3)
lines(c(1,8), c(8, 8), lwd=3)
lines(c(8,8), c(8, 1), lwd=3)
lines(c(8,7), c(1, 1), lwd=3)
lines(c(7,7), c(1, 0), lwd=3)
lines(c(7,0), c(0, 0), lwd=3)
rectan(c(1,9), width = 1.6, ht=.6, col = colors()[chip.col])
text(2, 9, "= domino piece", cex=2,adj=0)
#EndSection:
Failed Attempt to Solve the Mutilated Checkerboard Problem
Psych 355, Miyamoto, Spr '15

12. Failed Attempt at Solving the Mutilated Checkerboard Problem
Problem: Cover the mutilated checkerboard with domino pieces so that every domino covers two squares OR if this is impossible, explain why it is impossible.
Failure!
This is not a solution!
FACT: It is impossible to cover the mutilated checkerboard with dominoes.
Why is it impossible?
#Section:
plot.jm(c(-1, 9), c(-1, 10), axes=F)
j.dark <- colors()[82]
j.light <- 8
for (i in 1:4) for (j in 1:8) {
II <- (i - 1)*2 + .5
JJ <- j - .5
if (j %in% c(1,3,5,7)) j.col <- j.light else j.col <- j.dark
if (i != 1 | j != 8) rectan(c(JJ,II), width=1, ht=1, col=j.col) }
II <- (i - 1)*
if (j %in% c(2,4,6,8)) j.col <- j.light else j.col <- j.dark
if (i != 4 | j != 1) rectan(c(JJ,II), width=1, ht=1, col=j.col)
lines(c(0,0), c(0, 7), lwd=3)
lines(c(0,1), c(7, 7), lwd=3)
lines(c(1,1), c(7, 8), lwd=3)
lines(c(1,8), c(8, 8), lwd=3)
lines(c(8,8), c(8, 1), lwd=3)
lines(c(8,7), c(1, 1), lwd=3)
lines(c(7,7), c(1, 0), lwd=3)
lines(c(7,0), c(0, 0), lwd=3)
rectan(c(1,9), width = 1.6, ht=.6, col = colors()[chip.col])
text(2, 9, "= domino piece", cex=2,adj=0)
#EndSection:
Solution to the Mutilated Checkerboard Problem
Psych 355, Miyamoto, Spr '15

13. Solution to the Mutilated Checkerboard Problem
Problem: Cover the checkerboard with domino pieces so that every domino covers two squares OR if this is impossible, explain why it is impossible.
A Solution is Impossible!
A domino piece always covers one dark square and one light square. Therefore any solution covers an equal number of dark and light squares.
The mutilated checkerboard has 30 dark squares and 32 light squares so it is impossible to cover an equal number of dark and light squares.
Easy Version of the Mutilated Checkerboard Problem – The Matchmaker Problem
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14. Easy Version of the Mutilate Checkerboard Problem The Russian Marriage Problem (a.k.a. the Matchmaker Problem)
Hayes, 1978: [wording slightly altered below]
In a small Russian village, there were 32 bachelors and 32 unmarried women. A matchmaker arranges 32 highly satisfactory marriages. The village was happy and proud. One night, two bachelors got drunk and killed each other. Can the matchmaker come up with heterosexual marriages (one man, one woman) among the 62 survivors?
#Section:
plot.jm(c(-1, 9), c(-1, 10), axes=F)
j.dark <- 8 #colors()[82]
j.light <- 8
for (i in 1:4) for (j in 1:8) {
II <- (i - 1)*2 + .5
JJ <- j - .5
if (j %in% c(1,3,5,7)) j.col <- j.light else j.col <- j.dark
if (j %in% c(1,3,5,7)) j.tx <- "Woman" else j.tx <- "Man"
rectan(c(JJ,II), width=1, ht=1, col=j.col)
text(JJ, II, j.tx) }
II <- (i - 1)*
if (j %in% c(2,4,6,8)) j.col <- j.light else j.col <- j.dark
if (j %in% c(2,4,6,8)) j.tx <- "Woman" else j.tx <- "Man"
rectan(center=c(4,4), width=8, ht=8, lwd=3)
rectan(c(1* , ), width=1, ht=1, col="darkblue", density = 25, angle = 0)
rectan(c(1* , ), width=1, ht=1, col="darkblue", density = 25, angle = 0)
#EndSection:
There are 30 men and 32 women. Obviously there is no way to match them into a complete set of heterosexual couples.
Mutilated Checkerboard Problem & Russian Marriage Problem Are Isomorphs
Psych 355, Miyamoto, Spr '15

15. Mutilated Checkerboard Problem & Russian Marriage Problem
#Section:
plot.jm(c(-1, 9), c(-1, 10), axes=F)
j.dark <- 8 #colors()[82]
j.light <- 8
for (i in 1:4) for (j in 1:8) {
II <- (i - 1)*2 + .5
JJ <- j - .5
if (j %in% c(1,3,5,7)) j.col <- j.light else j.col <- j.dark
if (j %in% c(1,3,5,7)) j.tx <- "Woman" else j.tx <- "Man"
rectan(c(JJ,II), width=1, ht=1, col=j.col)
text(JJ, II, j.tx) }
II <- (i - 1)*
if (j %in% c(2,4,6,8)) j.col <- j.light else j.col <- j.dark
if (j %in% c(2,4,6,8)) j.tx <- "Woman" else j.tx <- "Man"
rectan(center=c(4,4), width=8, ht=8, lwd=3)
rectan(c(1* , ), width=1, ht=1, col="darkblue", density = 25, angle = 0)
rectan(c(1* , ), width=1, ht=1, col="darkblue", density = 25, angle = 0)
#EndSection:
The multilated checkerboard problem and the Russian marriage problem are problem isomorphs.
Problem Isomorphs: Problems that differ superficially but have identical logical structure.
Concept of Problem Isomorphs
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16. Concept of Problem Isomorphs
Problem isomorphs – structurally identical versions of a problem.
Basic fact about problem isomorphs: Some versions of a problem are harder to solve than other versions of the problem.
What is the psychological difference between the mutilated checkerboard problem and the matchmaker problem?
Kaplan and Simon: It is easier to solve the Russian marriage problem than the mutilated checkerboard problem, presumably because the Russian marriage version makes the importance of pairing men with women obvious. (See next slide)
Basic meaning of “morph” is “form” or “shape”.
Four Isomorphic Versions of the Mutilated Checkerboard Problem
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17. Kaplan & Simon: Four Isomorphic Versions of the Mutilated Checkerboard Problem
Blank board is hardest problem.
“Bread”/“Butter” word labels are easiest problem.
Colored & “Pink”/“Black” word labels are intermediate difficulty.
The salience of the pairing affects difficulty.
Blank (hardest)
Colored (intermediate)
See ‘e:\p355\rcode\mutilated checkerboard.doc’ for the R-code.
“Pink” & “Black” Word Labels (intermediate)
“Bread” & “Butter” (easiest)
Conclusions re Problem Representation
Psych 355, Miyamoto, Spr '15

18. Conclusion re Problem Representation
Some problem representations make problem solving easier than other problem representations.
Solving an insight problem often depends on finding a problem representation that make it obvious how to find the solution.
Examples that support these claims:
Mutilated checkerboard problem; Russian marriage problem; other isomorphic versions.
Circle problem. .
Cheap Necklace Problem – An Example of a False Constraint
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19. Cheap Necklace Problem (Chain Problem)
Cheap Necklace Problem: Convert these 4 strands of chains into a single loop by opening and closing only 3 links. (Insight problem)
This is an example of a problem that is difficult because people apply a false constraint to the problem representation.
Stop Here?
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20. Problem Definition for the Chain Problem
Initial state: 4 strands of chains, initially separated.
Goal state: One unbroken loop.
Moves: Open and close links.
Constraint: Only 3 links can be opened and closed.
Initial State
Goal State
What series of permissible moves will transform the initial state into the goal state?
Solution to the Chain Problem
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21. Solution to the Chain Problem
Open all three links of one strand.
Use these open links to link together the other three strands.
(Next – see how this would work)
Show How to Visualize the Solution
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22. Solution to the Chain Problem
Open all three links of one strand.
Use these open links to link together the other three strands.
Show how to visualize the solution
Psych 355, Miyamoto, Spr '15

23. Solution to the Chain Problem
Open all three links of one strand.
Use these open links to link together the other three strands.
Show how to visualize the solution
Psych 355, Miyamoto, Spr '15

24. Solution to the Chain Problem
Open all three links of one strand.
Use these open links to link together the other three strands.
Show how to visualize the solution
Psych 355, Miyamoto, Spr '15

25. Solution to the Chain Problem
Open all three links of one strand.
Use these open links to link together the other three strands.
Show how to visualize the solution
Psych 355, Miyamoto, Spr '15

26. Solution to the Chain Problem
Open all three links of one strand.
Use these open links to link together the other three strands.
Show how to visualize the solution
Psych 355, Miyamoto, Spr '15

27. Solution to the Chain Problem
Open all three links of one strand.
Use these open links to link together the other three strands.
Summary re Solution to the Cheap Necklace Problem
Psych 355, Miyamoto, Spr '15

28. Summary re Solution to the Chain Problem
Open all three links of one strand.
Use these open links to link together the other three strands.
Why is this solution hard to discover?
False constraint: People assume that they can only open the links at the ends of existing chains.
Often we have difficulty solving a problem because we add a requirement to the solution that is not a true requirement (false constraint).
Nine Dot Problem
Psych 355, Miyamoto, Spr '15

29. Nine-Dot Problem
Make a diagram that has 9 dots as shown below. Draw 4 straight lines that connect all of the dots without lifting the pencil or pen from the paper.
The Nine-Dot Problem is difficult because people tend to assume a false constraint. (Same difficulty as with the Cheap Necklace Problem.)
Failed Attempt at a Solution to the Nine-Dot Problem
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30. Nine-Dot Problem (cont.)
Dead-end thinking. This is NOT a solution (5 lines are used).
Solution to the Nine-Dot Problem
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31. Solution to the Nine-Dot Problem
"Thinking inside the box" – People impose constraints on the problem that aren't there. To solve this problem, you have to “think outside the box.”
False constraint: In a failed solution, people assume that they must stay within the boundaries of the square.
It can be useful to "think outside the box" – discard false constraints on the problem solution.
So Far: Two Obstacles to Problem Solving
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32. Tuesday, May 26, 2015: The Lecture Ended Here
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33. So Far: Two Obstacles to Problem Solving
Obstacle #1: A poor initial problem representation makes it difficult to solve a problem.
Example: The Circle Problem
Example: The Mutilated Checkerboard Problem
Remedy: Change the problem representation (sometimes a radical change is helpful.)
Obstacle #2: People sometimes place a false constraint on the permissible ways to solve the problem.
Example: Cheap Necklace Problem.
Example: Nine-Dot Problem
Remedy: Examine the constraints – are you imposing a false constraint?
END: Time Permitting, a Fishing Story
Psych 355, Miyamoto, Spr '15

34. Time Permitting: An Example of a Real False Constraint – A Fishing Story
Time permitting, give practical example of a false constraint.
JM was stuck on a rock in the middle of a deep rapid river (Middle Fork of the Snoqualmie River).
Problem: How to get from the rock to the shore (alive)?
False Constraint: JM only considered routes through the rapids that would get him to the shore dry and alive. These routes were all very dangerous.
Solution: Choose a route that would get JM to the shore wet but alive.
END
Psych 355, Miyamoto, Spr '15